isl_affine_hull.c

   1 #include "isl_ctx.h"
   2 #include "isl_seq.h"
   3 #include "isl_set.h"
   4 #include "isl_lp.h"
   5 #include "isl_map.h"
   6 #include "isl_map_private.h"
   7 #include "isl_equalities.h"
   8 #include "isl_sample.h"
   9
  10 struct isl_basic_map *isl_basic_map_implicit_equalities(
  11                                                 struct isl_basic_map *bmap)
  12 {
  13         int i;
  14         int rational;
  15         isl_int opt;
  16         isl_int opt_denom;
  17         struct isl_ctx *ctx;
  18
  19         if (!bmap)
  20                 return bmap;
  21
  22         if (F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
  23                 return bmap;
  24         if (F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))
  25                 return bmap;
  26
  27         ctx = bmap->ctx;
  28         rational = F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
  29         isl_int_init(opt);
  30         isl_int_init(opt_denom);
  31         if (!rational)
  32                 isl_int_set_si(opt_denom, 1);
  33         for (i = 0; i < bmap->n_ineq; ++i) {
  34                 enum isl_lp_result res;
  35                 res = isl_solve_lp(bmap, 1, bmap->ineq[i]+1, ctx->one,
  36                                         &opt, rational ? &opt_denom : NULL);
  37                 if (res == isl_lp_unbounded)
  38                         continue;
  39                 if (res == isl_lp_error)
  40                         goto error;
  41                 if (res == isl_lp_empty) {
  42                         bmap = isl_basic_map_set_to_empty(bmap);
  43                         break;
  44                 }
  45                 if (!isl_int_is_one(opt_denom))
  46                         continue;
  47                 isl_int_add(opt, opt, bmap->ineq[i][0]);
  48                 if (isl_int_is_zero(opt)) {
  49                         isl_basic_map_inequality_to_equality(bmap, i);
  50                         --i;
  51                 }
  52         }
  53         isl_int_clear(opt_denom);
  54         isl_int_clear(opt);
  55
  56         F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
  57         return bmap;
  58 error:
  59         isl_int_clear(opt);
  60         isl_basic_map_free(bmap);
  61         return NULL;
  62 }
  63
  64 /* Make eq[row][col] of both bmaps equal so we can add the row
  65  * add the column to the common matrix.
  66  * Note that because of the echelon form, the columns of row row
  67  * after column col are zero.
  68  */
  69 static void set_common_multiple(
  70         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  71         unsigned row, unsigned col)
  72 {
  73         isl_int m, c;
  74
  75         if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
  76                 return;
  77
  78         isl_int_init(c);
  79         isl_int_init(m);
  80         isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
  81         isl_int_divexact(c, m, bset1->eq[row][col]);
  82         isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
  83         isl_int_divexact(c, m, bset2->eq[row][col]);
  84         isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
  85         isl_int_clear(c);
  86         isl_int_clear(m);
  87 }
  88
  89 /* Delete a given equality, moving all the following equalities one up.
  90  */
  91 static void delete_row(struct isl_basic_set *bset, unsigned row)
  92 {
  93         isl_int *t;
  94         int r;
  95
  96         t = bset->eq[row];
  97         bset->n_eq--;
  98         for (r = row; r < bset->n_eq; ++r)
  99                 bset->eq[r] = bset->eq[r+1];
 100         bset->eq[bset->n_eq] = t;
 101 }
 102
 103 /* Make first row entries in column col of bset1 identical to
 104  * those of bset2, using the fact that entry bset1->eq[row][col]=a
 105  * is non-zero.  Initially, these elements of bset1 are all zero.
 106  * For each row i < row, we set
 107  *              A[i] = a * A[i] + B[i][col] * A[row]
 108  *              B[i] = a * B[i]
 109  * so that
 110  *              A[i][col] = B[i][col] = a * old(B[i][col])
 111  */
 112 static void construct_column(
 113         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 114         unsigned row, unsigned col)
 115 {
 116         int r;
 117         isl_int a;
 118         isl_int b;
 119         unsigned total;
 120
 121         isl_int_init(a);
 122         isl_int_init(b);
 123         total = 1 + bset1->dim;
 124         for (r = 0; r < row; ++r) {
 125                 if (isl_int_is_zero(bset2->eq[r][col]))
 126                         continue;
 127                 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
 128                 isl_int_divexact(a, bset1->eq[row][col], b);
 129                 isl_int_divexact(b, bset2->eq[r][col], b);
 130                 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
 131                                               b, bset1->eq[row], total);
 132                 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
 133         }
 134         isl_int_clear(a);
 135         isl_int_clear(b);
 136         delete_row(bset1, row);
 137 }
 138
 139 /* Make first row entries in column col of bset1 identical to
 140  * those of bset2, using only these entries of the two matrices.
 141  * Let t be the last row with different entries.
 142  * For each row i < t, we set
 143  *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 144  *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 145  * so that
 146  *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 147  */
 148 static int transform_column(
 149         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 150         unsigned row, unsigned col)
 151 {
 152         int i, t;
 153         isl_int a, b, g;
 154         unsigned total;
 155
 156         for (t = row-1; t >= 0; --t)
 157                 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
 158                         break;
 159         if (t < 0)
 160                 return 0;
 161
 162         total = 1 + bset1->dim;
 163         isl_int_init(a);
 164         isl_int_init(b);
 165         isl_int_init(g);
 166         isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
 167         for (i = 0; i < t; ++i) {
 168                 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
 169                 isl_int_gcd(g, a, b);
 170                 isl_int_divexact(a, a, g);
 171                 isl_int_divexact(g, b, g);
 172                 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
 173                                 total);
 174                 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
 175                                 total);
 176         }
 177         isl_int_clear(a);
 178         isl_int_clear(b);
 179         isl_int_clear(g);
 180         delete_row(bset1, t);
 181         delete_row(bset2, t);
 182         return 1;
 183 }
 184
 185 /* The implementation is based on Section 5.2 of Michael Karr,
 186  * "Affine Relationships Among Variables of a Program",
 187  * except that the echelon form we use starts from the last column
 188  * and that we are dealing with integer coefficients.
 189  */
 190 static struct isl_basic_set *affine_hull(
 191         struct isl_basic_set *bset1, struct isl_basic_set *bset2)
 192 {
 193         unsigned total;
 194         int col;
 195         int row;
 196
 197         total = 1 + bset1->dim;
 198
 199         row = 0;
 200         for (col = total-1; col >= 0; --col) {
 201                 int is_zero1 = row >= bset1->n_eq ||
 202                         isl_int_is_zero(bset1->eq[row][col]);
 203                 int is_zero2 = row >= bset2->n_eq ||
 204                         isl_int_is_zero(bset2->eq[row][col]);
 205                 if (!is_zero1 && !is_zero2) {
 206                         set_common_multiple(bset1, bset2, row, col);
 207                         ++row;
 208                 } else if (!is_zero1 && is_zero2) {
 209                         construct_column(bset1, bset2, row, col);
 210                 } else if (is_zero1 && !is_zero2) {
 211                         construct_column(bset2, bset1, row, col);
 212                 } else {
 213                         if (transform_column(bset1, bset2, row, col))
 214                                 --row;
 215                 }
 216         }
 217         isl_basic_set_free(bset2);
 218         isl_assert(ctx, row == bset1->n_eq, goto error);
 219         return bset1;
 220 error:
 221         isl_basic_set_free(bset1);
 222         return NULL;
 223 }
 224
 225 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_ctx *ctx,
 226         struct isl_vec *vec)
 227 {
 228         int i;
 229         int k;
 230         struct isl_basic_set *bset = NULL;
 231
 232         if (!vec)
 233                 return NULL;
 234         isl_assert(ctx, vec->size != 0, goto error);
 235
 236         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
 237         for (i = bset->dim - 1; i >= 0; --i) {
 238                 k = isl_basic_set_alloc_equality(bset);
 239                 if (k < 0)
 240                         goto error;
 241                 isl_seq_clr(bset->eq[k], 1 + bset->dim);
 242                 isl_int_neg(bset->eq[k][0], vec->block.data[1 + i]);
 243                 isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
 244         }
 245         isl_vec_free(ctx, vec);
 246
 247         return bset;
 248 error:
 249         isl_basic_set_free(bset);
 250         isl_vec_free(ctx, vec);
 251         return NULL;
 252 }
 253
 254 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
 255         struct isl_basic_set *bset, isl_int *eq, int up)
 256 {
 257         struct isl_basic_set *slice = NULL;
 258         struct isl_vec *sample;
 259         struct isl_basic_set *point;
 260         int k;
 261
 262         slice = isl_basic_set_copy(bset);
 263         if (!slice)
 264                 goto error;
 265         slice = isl_basic_set_extend(slice, 0, slice->dim, 0, 1, 0);
 266         k = isl_basic_set_alloc_equality(slice);
 267         if (k < 0)
 268                 goto error;
 269         isl_seq_cpy(slice->eq[k], eq, 1 + slice->dim);
 270         if (up)
 271                 isl_int_add_ui(slice->eq[k][0], slice->eq[k][0], 1);
 272         else
 273                 isl_int_sub_ui(slice->eq[k][0], slice->eq[k][0], 1);
 274
 275         sample = isl_basic_set_sample(slice);
 276         if (!sample)
 277                 goto error;
 278         if (sample->size == 0) {
 279                 isl_vec_free(ctx, sample);
 280                 point = isl_basic_set_empty(ctx, 0, bset->dim);
 281         } else
 282                 point = isl_basic_set_from_vec(ctx, sample);
 283
 284         return point;
 285 error:
 286         isl_basic_set_free(slice);
 287         return NULL;
 288 }
 289
 290 /* After computing the rational affine hull (by detecting the implicit
 291  * equalities), we remove all equalities found so far, compute
 292  * the integer affine hull of what is left, and then add the original
 293  * equalities back in.
 294  *
 295  * The integer affine hull is constructed by successively looking
 296  * a point that is affinely independent of the points found so far.
 297  * In particular, for each equality satisfied by the points so far,
 298  * we check if there is any point on the corresponding hyperplane
 299  * shifted by one (in either direction).
 300  */
 301 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 302 {
 303         int i, j;
 304         struct isl_mat *T2 = NULL;
 305         struct isl_basic_set *bset = NULL;
 306         struct isl_basic_set *hull = NULL;
 307         struct isl_vec *sample;
 308         struct isl_ctx *ctx;
 309
 310         bmap = isl_basic_map_implicit_equalities(bmap);
 311         if (!bmap)
 312                 return NULL;
 313         ctx = bmap->ctx;
 314         if (bmap->n_ineq == 0)
 315                 return bmap;
 316
 317         if (F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
 318                 bmap = isl_basic_map_cow(bmap);
 319                 isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 320                 return bmap;
 321         }
 322
 323         bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
 324         bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
 325
 326         sample = isl_basic_set_sample(isl_basic_set_copy(bset));
 327         hull = isl_basic_set_from_vec(ctx, sample);
 328
 329         for (i = 0; i < bset->dim; ++i) {
 330                 struct isl_basic_set *point;
 331                 for (j = 0; j < hull->n_eq; ++j) {
 332                         point = outside_point(ctx, bset, hull->eq[j], 1);
 333                         if (!point)
 334                                 goto error;
 335                         if (!F_ISSET(point, ISL_BASIC_SET_EMPTY))
 336                                 break;
 337                         isl_basic_set_free(point);
 338                         point = outside_point(ctx, bset, hull->eq[j], 0);
 339                         if (!point)
 340                                 goto error;
 341                         if (!F_ISSET(point, ISL_BASIC_SET_EMPTY))
 342                                 break;
 343                         isl_basic_set_free(point);
 344                 }
 345                 if (j == hull->n_eq)
 346                         break;
 347                 hull = affine_hull(hull, point);
 348         }
 349
 350         isl_basic_set_free(bset);
 351         bset = NULL;
 352         bmap = isl_basic_map_cow(bmap);
 353         if (!bmap)
 354                 goto error;
 355         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 356         if (T2)
 357                 hull = isl_basic_set_preimage(ctx, hull, T2);
 358         bmap = isl_basic_map_intersect(bmap,
 359                                         isl_basic_map_overlying_set(hull,
 360                                                 isl_basic_map_copy(bmap)));
 361
 362         return isl_basic_map_finalize(bmap);
 363 error:
 364         isl_mat_free(ctx, T2);
 365         isl_basic_set_free(bset);
 366         isl_basic_set_free(hull);
 367         isl_basic_map_free(bmap);
 368         return NULL;
 369 }
 370
 371 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 372 {
 373         return (struct isl_basic_set *)
 374                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 375 }
 376
 377 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 378 {
 379         int i;
 380         struct isl_basic_map *model = NULL;
 381         struct isl_basic_map *hull = NULL;
 382         struct isl_set *set;
 383
 384         if (!map)
 385                 return NULL;
 386
 387         if (map->n == 0) {
 388                 hull = isl_basic_map_empty(map->ctx,
 389                                         map->nparam, map->n_in, map->n_out);
 390                 isl_map_free(map);
 391                 return hull;
 392         }
 393
 394         map = isl_map_align_divs(map);
 395         model = isl_basic_map_copy(map->p[0]);
 396         set = isl_map_underlying_set(map);
 397         set = isl_set_cow(set);
 398         if (!set)
 399                 goto error;
 400
 401         for (i = 0; i < set->n; ++i) {
 402                 set->p[i] = isl_basic_set_cow(set->p[i]);
 403                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 404                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 405                 if (!set->p[i])
 406                         goto error;
 407         }
 408         set = isl_set_remove_empty_parts(set);
 409         if (set->n == 0) {
 410                 hull = isl_basic_map_empty(set->ctx,
 411                                     model->nparam, model->n_in, model->n_out);
 412                 isl_basic_map_free(model);
 413         } else {
 414                 struct isl_basic_set *bset;
 415                 while (set->n > 1) {
 416                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 417                         if (!set->p[0])
 418                                 goto error;
 419                 }
 420                 bset = isl_basic_set_copy(set->p[0]);
 421                 hull = isl_basic_map_overlying_set(bset, model);
 422         }
 423         isl_set_free(set);
 424         hull = isl_basic_map_simplify(hull);
 425         return isl_basic_map_finalize(hull);
 426 error:
 427         isl_basic_map_free(model);
 428         isl_set_free(set);
 429         return NULL;
 430 }
 431
 432 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 433 {
 434         return (struct isl_basic_set *)
 435                 isl_map_affine_hull((struct isl_map *)set);
 436 }